Exact Schema Theory and Markov Chain Models for Genetic Programming and Variable-length Genetic Algorithms with Homologous Crossover

Poli, Riccardo; McPhee, Nicholas Freitag; Rowe, Jonathan E.

doi:10.1023/b:genp.0000017010.41337.a7

Cited by 70 publications

(90 citation statements)

References 33 publications

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“…Theoretical work concerning GP has always been undertaken since its early days [6], the majority of which has applied schema theory [4,17]. Schema theories are based on the idea of partitioning the search space into subsets, called schemata, and modelling the behaviour and dynamics of the population over the schemata.…”

Section: Introductionmentioning

confidence: 99%

On the Analysis of Simple Genetic Programming for Evolving Boolean Functions

Mambrini¹,

Oliveto²

2016

Lecture Notes in Computer Science

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Abstract. This work presents a first step towards a systematic time and space complexity analysis of genetic programming (GP) for evolving functions with desired input/output behaviour. Two simple GP algorithms, called (1+1) GP and (1+1) GP*, equipped with minimal function (F) and terminal (L) sets are considered for evolving two standard classes of Boolean functions. It is rigorously proved that both algorithms are efficient for the easy problem of evolving conjunctions of Boolean variables with the minimal sets. However, if an extra function (i.e. NOT) is added to F, then the algorithms require at least exponential time to evolve the conjunction of n variables. On the other hand, it is proved that both algorithms fail at evolving the difficult parity function in polynomial time with probability at least exponentially close to 1. Concerning generalisation, it is shown how the quality of the evolved conjunctions depends on the size of the training set s while the evolved exclusive disjunctions generalize equally badly independent of s.

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Section: Introductionmentioning

confidence: 99%

On the Analysis of Simple Genetic Programming for Evolving Boolean Functions

Mambrini¹,

Oliveto²

2016

Lecture Notes in Computer Science

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“…However, recently, the first exact GP schema theories have become available (Poli 2001), which give exact formulations for the expected number of individuals sampling a schema at the next generation. Initially, these exact theories were only applicable to GP with one-point crossover (see > Sect.…”

Section: Schema Theoriesmentioning

confidence: 99%

“…While these results have been extended to more general search spaces (Rudolph 1996) and, as it is discussed in > Sect. 4.2, Markov chain models of some forms of GP have been recently derived (e.g., see Poli et al (2004)), the calculations involved in setting up and studying these models are of considerable mathematical complexity. Furthermore, extending the work to infinite search spaces (which is required to model the traditional form of GP where operators can create trees of potentially any size over a number of generations) presents even bigger challenges.…”

Section: Is Gp Guaranteed To Find Solutions To Problems?mentioning

confidence: 99%

Genetic Programming — Introduction, Applications, Theory and Open Issues

Vanneschi

Poli

2012

Handbook of Natural Computing

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“…These, in their turn, have allowed for a reconciliation of previously, seemingly antagonistic formulations of the dynamics of GAs. Such coarse grained formulations have further been extended to variable-length linear representations and tree representations [7,9,8,5,10,11,17], thus leading to a unification of the theory underlying GAs and GP. Given that coarse grained formulations have led to a great number of advances in the theory of GAs and GP, and that in the case of fixed-length strings a natural coarse graining can be implemented via a coordinate transformation, it is natural to ask if such basis transformations also exist in the more complicated cases of variable-length strings and trees.…”

Section: Introductionmentioning

confidence: 99%